"In mathematics and physics, vector is a term that refers colloquially to some quantities that cannot be expressed by a single number (a scalar), or to elements of some vector spaces."
Vectors in mathematics are quantities that have both a magnitude (or length) and a direction.
Definition of Vectors: Vectors are quantities that have both magnitude and direction. They are used to describe physical quantities such as velocity, acceleration, force, and displacement.
Vector Addition: This operation involves combining two or more vectors to obtain a resultant vector. It is achieved by adding the corresponding components of the individual vectors.
Vector Subtraction: This is the process of finding the difference between two vectors. It is achieved by subtracting the corresponding components of the individual vectors.
Scalar Multiplication of Vectors: This is an operation in which a vector is multiplied by a scalar (a real number). The magnitude of the resulting vector is the product of the scalar and the magnitude of the original vector.
Dot Product: This is an operation that involves multiplying the corresponding components of two vectors and then adding up the products. The result is a scalar quantity.
Cross Product: This is an operation that involves finding a vector that is orthogonal (perpendicular) to two given vectors. The magnitude of the resulting vector is the product of the magnitudes of the two vectors multiplied by the sine of the angle between them.
Vector Properties: There are several important properties of vectors. These include commutativity, associativity, distributivity, and the fact that the magnitude of a vector is always non-negative.
Unit Vectors: These are vectors with a magnitude of 1. They are often used to express directions.
Vector Components: The components of a vector are the projections of the vector onto the axes of a coordinate system. They can be used to calculate the magnitude and direction of the vector.
Vector Projection: This is the process of finding the component of a vector that lies along a given direction.
Vector Decomposition: This is the process of expressing a vector as the sum of two or more vectors.
Applications of Vectors: Vectors are used in a wide variety of applications, including motion in two and three dimensions, statics, fluid mechanics, electromagnetism, and more.
Position vector: A vector that represents the position of a point in space, relative to a chosen reference point or origin.
Velocity vector: A vector that represents the velocity of an object, i.e., the rate and direction of its motion.
Acceleration vector: A vector that represents the acceleration of an object, i.e., the rate and direction of change in its velocity.
Force vector: A vector that represents a force acting on an object, i.e., the magnitude and direction of the force.
Weight vector: A vector that represents the weight of an object, i.e., the force of gravity acting on it.
Displacement vector: A vector that represents the change in position of an object, i.e., the difference between its final and initial position.
Momentum vector: A vector that represents the momentum of an object, i.e., the product of its mass and velocity.
Torque vector: A vector that represents the torque acting on an object, i.e., the product of the force and the distance from the axis of rotation.
Angular momentum vector: A vector that represents the angular momentum of an object, i.e., the product of its moment of inertia and angular velocity.
Magnetic field vector: A vector that represents the direction and strength of a magnetic field.
Electric field vector: A vector that represents the direction and strength of an electric field.
Gravitational field vector: A vector that represents the direction and strength of a gravitational field.
Stress vector: A vector that represents the force per unit area acting on a material or object.
Strain vector: A vector that represents the change in shape or size of a material or object due to stress.
Polarization vector: A vector that represents the direction and magnitude of polarization of a medium or substance.
Spin vector: A vector that represents the intrinsic angular momentum of a particle or system.
"Historically, vectors were introduced in geometry and physics (typically in mechanics) for quantities that have both a magnitude and a direction, such as displacements, forces, and velocity."
"Such quantities are represented by geometric vectors in the same way as distances, masses, and time are represented by real numbers."
"The term vector is also used, in some contexts, for tuples, which are finite sequences of numbers of a fixed length."
"Both geometric vectors and tuples can be added and scaled."
"These vector operations led to the concept of a vector space, which is a set equipped with a vector addition and a scalar multiplication that satisfy some axioms generalizing the main properties of operations on the above sorts of vectors."
"A vector space formed by geometric vectors is called a Euclidean vector space."
"A vector space formed by tuples is called a coordinate vector space."
"Many vector spaces are considered in mathematics, such as extension field, polynomial rings, algebras, and function spaces."
"The term vector is generally not used for elements of these vector spaces..."
"...and is generally reserved for geometric vectors, tuples, and elements of unspecified vector spaces..."
"a term that refers colloquially to some quantities that cannot be expressed by a single number (a scalar)..."
"...quantities that have both a magnitude and a direction, such as displacements, forces, and velocity."
"Both geometric vectors and tuples can be added and scaled..."
"...a vector addition and a scalar multiplication that satisfy some axioms generalizing the main properties of operations on the above sorts of vectors."
"A vector space formed by geometric vectors is called a Euclidean vector space."
"A vector space formed by tuples is called a coordinate vector space."
"...extension field, polynomial rings, algebras, and function spaces."
"The term vector is generally not used for elements of these vector spaces..."
"...and is generally reserved for geometric vectors, tuples, and elements of unspecified vector spaces."